The first step in deriving the discontinuous Galerkin Poisson solver consists in multiplying the Poisson equation (REF ) by a test function \(v\in V_h^k\) , integrating over each cell \(I_i = [x_i,x_{i+1}]\)  and summing up all terms. For more details, see [1]}. This gives, in 1d, the following equation
\(\begin{split}\sum _{i=0}^{N-1} \bigg ( \int _{I_i}&\phi ^{\prime }(x)v^{\prime }(x)\,dx - [\phi ^{\prime }(x_{i})]\lbrace v(x_i)\rbrace  - \\&\lbrace \phi ^{\prime }(x_{i})\rbrace [v(x_i)] + \frac{\sigma }{h}[v(x_n)][\phi (x_n)] \bigg ) \\ &= \int _{\Omega } (1-\rho (x))v(x)\,dx,\end{split}\) 
